Which of the following is an odd function? f(x) = x3 + 5x2 + x mc023-1.jpg f(x) = x2 + x f(x) = –x

Given the functions
(a) f(x) = x³ + 5x² + x
(b) f(x) = x² + x
(c) f(x) = -x

Function (a)
f(-x) = (-x)³ + 5(-x)² + (-x)
       = -x³ + 5x² - x
       = -(x³ - 5x² + x)
The function is neither even nor odd.

Function (b)
f(-x) = (-x)² + (-x)
        = -(-x² + x)
The function is neither even nor odd.

Function (c)
f(-x) = -(-x)
       = x
       = -f(x)
Because f(-x) = -f(x) the function is odd.

Answer: f(x) = -x is an odd function.